The methods for solving the diffusion equation were presented
for cases of fixed boundary conditions.
However, there many examples of kinetic processes in materials
where boundaries (e.g. interfaces, phase boundaries) move in
response or because of diffusion.
Below, methods to treat such problems will be shown to be straightforward
extensions of the diffusion equation-the additional physics is a
conservation
principle relating the velocity of the moving interface the rate at which a
conserved quantity is consumed per unit area of the interface.
While exact solutions are difficult to obtain, a few general results and
approximations
can be obtained and applied to materials processes.

The analysis of the moving interface problem originates with Stefan who
was developing a model for the rate of melting of the polar ice-caps and
icebergs. This problem remains as one of the biggest alloy solidification
problems. Heat must be conducted from the oceans to the melting interface
to to provide the latent heat of melting and salt must be supplied as well
since the equilibrium concentrations salt in the liquid and solid differ.

Interface Motion due to Heat Absorption at the Interface

To simplify the analysis of the problem, consider the heat-flux
problem independently; specifically,
consider freezing a liquid-solid mixture by extraction of heat:

Figure 26-1:
Schematic illustration
of a temperature distribution resulting in the freezing and motion of
the liquid/solid interface.
The velocity of the interface will depend on the difference
in enthalpy density in each phase.

Assuming density, , is same in each phase, and equating the
volume swept out with heat required for the phase change:

(26-1)

where is the enthalpy per unit volume, therefore

(26-2)

Equation 26-2 is known as the ``Stefan Condition,'' is
the position of the (assumed planar) interface.

It is probably wise to check for wayward minus signs.
Consider the usual case,
,
and suppose the thermal diffusivity in the solid phase is zero (i.e. all
heat is
absorbed by the interface and supplied by the liquid reservoir); does the
velocity of
the interface have the expected sign?

Therefore the thermal diffusion equations become:

(26-3)

(26-4)

with the new unknown function, the interface position ,
to be determined by the subsidiary Stefan condition:

(26-5)

Mass Diffusion in an Alloy

The Stefan condition relates the velocity of the interface to
the ``jump'' in the density of an extensive quantity.
For the case of heat above, that quantity was the enthalpy
density.
Next, the diffusion of chemical species will be coupled to
the jump in alloy composition (amount/volume) at a moving
interface--an analogous Stefan condition results.

Consider a diffusion couple between two alloys at different compositions
for a system with multiple phases in equilibrium at a given temperature.

Figure 26-2:
Schematic illustration of
diffusion in an alloy with more than one equilibrium phase at a given
temperature

The mass balance at the moving interface is related to the
phase diagram:

Figure 26-3:
Illustration
of the composition difference at an interface in local equilibrium.

(26-6)

This must be balance by the amount going in:

(26-7)

minus the amount going out

(26-8)

Therefore, the Stefan condition is:

(26-9)

Simple Stefan Example

A limiting case for the mass diffusion case is developed below; the
result that the interface grows as is derived.
This result, as shown in the textbook, is a general one for
the Stefan problem with uniform diffusivity in each phase. Therefore,
this result is applicable to materials processes where material must
diffuse through a growing phase towards an interface where is can react
and form new material--such as oxidation of a surface.

The coupled diffusion equations are:

(26-10)

With the simplifying assumptions that
and
a steady-state profile applies in -phase, the concentration profiles
become:

(26-11)

Incorporating this limiting case into the Stefan condition and integrating,

(26-12)

Morphological Instabilities

A growth interface can undergo a morphological instability in cases
when the driving force for growth (or transformation) is very large.
The commonly observed example is that of a snowflake--which is a
beautiful structure, but from simple considerations may appear to
have much more surface energy than one might expect.
In fact, the surface energy `competes' with the driving force
for transformation--as the driving force increases, the amount
of `extra' surface of the growth shape increases.
On the other hand, if surface tension is very large compared to the
volumetric driving force then the tendency for an interface
to become unstable is decreases.

Instability of a Pure Liquid-Solid Interface

Consider the solidification of a pure liquid above its
melting point by removing heat through a walls which are
kept at a fixed temperature.

In this case, solidification begins at the walls and the
solidification interface moves towards the center of the
container at a rate which is dictated by how fast the
latent heat of solidification can be conducted through
the freshly grown solid phase and out through the walls.
In this case, the interface is completely stable and
the interface moves stably until all the liquid disappears.

Now consider the solidification of a pure liquid which
has been carefully supercooled below its melting point
with no nucleation.
If the solid phase is nucleated by a seed at the center
of the container, then solidification proceeds as heat
is conducted to the supercooled liquid and through the
container walls.

Figure 26-5:
Illustration of the solidification of a supercooled
liquid.

If effects of gravity are eliminated, then such an
experiment can be carried out with only thermal
diffusion through the liquid phase and no convection.
In this case, the interface is unstable and any small
undulations in the surface can grow into dendrites.

The essential difference between Figure 26-4
and Figure 26-5 is that in the unstable case
the new phase is growing into an unstable phase.
The basic idea can be described in fairly simple terms.
The supercooled liquid conducts heat which is generated by
solidification;
when a small protuberance forms at the interface, it pokes into
liquid at a slightly lower temperature which can more efficiently
conduct heat and therefore the protuberance continues to grow.

Alloy Solidification

A typical casting microstructure has a morphological instability:

Figure 26-6:
Typical casting microstructures.
The micrograph on the left comes from near the container surface.
If you are viewing in HTML, click on figure to see a phase field
simulation of grain growth computed by J.A. Warren of NIST.

This is a puzzle: The morphological instability occurs
for the case illustrated in Fig. 26-4--which
is the case that was argued to be stable.

Constitutional Supercooling

The puzzle is solved by showing that the liquid near the growing
interface is made unstable by composition variations due to
the limited rate of mass diffusion.
In this case, the instability is due to composition and not temperature.

The analogy between the thermal instability of a pure
substance and the instability of alloy at constant temperature
can be understood by referring to an isothermal
line in a binary phase diagram.

Figure 26-7:
Illustration of part of a binary phase diagram.
If the solid phase is growing into an unstable liquid at and
composition (
) then the interface will be unstable.

For a solid growing into a liquid phase, the advancing solid must reject
solute into the liquid phase.
The rate of advance is limited by the rate at which rejected solute can
be diffused away, just as in the thermal case where interface motion
is limited by the rate at which heat is diffused away.

Suppose that a material with a uniform composition, in
Fig. 26-7, is uniformly quenched into the
two-phase region.
The liquid is effectively under-cooled; such a system is
called constitutionally under-cooled.
Thus, a solidification front which starts from the edges of the
container will become unstable for the same reasons that the
front in
26-5
is unstable.

Mullins-Sekerka Instability

Both the constitutional supercooling and the thermal undercooling
interfaces were analyzed by Mullins and Sekerka.
They were able to determine a relationship between the wavelength
of the instability, the surface tension, the transport coefficients,
and the driving forces.

The analysis begins by introducing a dimensionless variable for
temperature in one case and composition in the other:

(26-13)

The interface condition is related to the curvature through the
Gibbs-Thompson effect:

(26-14)

where
is a capillary length:

(26-15)

This can be inserted into a set of moving interface diffusion equations
and the stability of the interface can be evaluated by perturbation
analysis.

All perturbation wavelengths greater that
can grow:

(26-16)

where is the ratio of solid to liquid transport coefficients and
is an effective diffusion length given by the interface-controlling
diffusivity divided
by the velocity of the interface.

The fastest growing wavelength is given by

(26-17)

It is expected that
will determining the scale
of the resulting microstructure.